Problem 46 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$y=\frac{2 x+3}{x-4} ; \quad(3,9),\left(0,-\frac{3}{4}\right),(5,13)$$

Step-by-Step Solution

Answer

The ordered pairs \( \left(0, -\frac{3}{4}\right) \) and (5, 13) satisfy the equation.

Step 1 - Write Down the Equation

The given equation is \[y = \frac{2x+3}{x-4}\]

Step 2 - Test the First Ordered Pair (3,9)

Substitute \(x = 3\) and \(y = 9\) into the equation:\[9 = \frac{2(3)+3}{3-4}\]Simplify the right side:\[9 = \frac{6+3}{-1}\]\[9 = \frac{9}{-1}\]\[9 = -9\]Since 9 does not equal -9, the ordered pair (3,9) does not satisfy the equation.

Step 3 - Test the Second Ordered Pair \( \left(0, -\frac{3}{4}\right) \)

Substitute \(x = 0\) and \(y = -\frac{3}{4}\) into the equation:\[-\frac{3}{4} = \frac{2(0)+3}{0-4}\]Simplify the right side:\[-\frac{3}{4} = \frac{3}{-4}\]\[-\frac{3}{4} = -\frac{3}{4}\]Since both sides are equal, the ordered pair \( \left(0, -\frac{3}{4}\right) \) satisfies the equation.

Step 4 - Test the Third Ordered Pair (5,13)

Substitute \(x = 5\) and \(y = 13\) into the equation:\[13 = \frac{2(5)+3}{5-4}\]Simplify the right side:\[13 = \frac{10+3}{1}\]\[13 = 13\]Since both sides are equal, the ordered pair (5,13) satisfies the equation.

Key Concepts

ordered pairs

Ordered pairs are fundamental in algebra and coordinate geometry. They consist of two elements, usually written in the form \( (x, y) \), where x represents the horizontal coordinate and y represents the vertical coordinate. Each ordered pair corresponds to a specific point on the coordinate plane. When solving algebraic equations, ordered pairs are used to determine if the pair satisfies the equation by substituting the values of x and y.

substituting values

Substituting values is a technique used to determine if a specific ordered pair is a solution to an algebraic equation. The process involves replacing the variables in the equation with the given values from the ordered pair. For instance, in the equation \( y = \frac{2x+3}{x-4} \), if we want to test the pair \( (3, 9) \), we substitute \( x = 3 \) and \( y = 9 \) into the equation. Then, we simplify the expression to see if both sides of the equation are equal.

algebraic equations

Algebraic equations represent relationships between variables. In our given example, the equation \( y = \frac{2x+3}{x-4} \) shows how y changes with respect to x. To understand if a pair \( (x, y) \) satisfies the equation, we substitute the values of x and y into the equation. If the left side equals the right side after simplifying, then that ordered pair is a solution to the equation. Each algebraic equation can have multiple solutions, a single solution, or no solutions. Testing each given ordered pair helps us identify which ones are solutions.

simplifying expressions

Simplifying expressions involves breaking down complex mathematical expressions into simpler forms. This process is crucial when testing ordered pairs in algebraic equations. For example, after substituting values from the pair \( (0, -\frac{3}{4}) \), the equation \( -\frac{3}{4} = \frac{2(0) + 3}{0 - 4} \) simplifies to \( -\frac{3}{4} = -\frac{3}{4} \). By simplifying, we check if both sides of the equation are equal, confirming if the ordered pair satisfies the algebraic equation. Simplifying helps us see the equivalence or discrepancy between the two sides of an equation clearly.